Indians predated Newton 'discovery' by 250 years Options

[PB.]

Source: http://www.physorg.com/news106238636.html
Their source: University of Manchester

A little known school of scholars in southwest India discovered one of the founding principles of modern mathematics hundreds of years before Newton according to new research.

Dr George Gheverghese Joseph from The University of Manchester says the 'Kerala School' identified the 'infinite series'- one of the basic components of calculus - in about 1350.

The discovery is currently - and wrongly - attributed in books to Sir Isaac Newton and Gottfried Leibnitz at the end of the seventeenth centuries.

The team from the Universities of Manchester and Exeter reveal the Kerala School also discovered what amounted to the Pi series and used it to calculate Pi correct to 9, 10 and later 17 decimal places.

And there is strong circumstantial evidence that the Indians passed on their discoveries to mathematically knowledgeable Jesuit missionaries who visited India during the fifteenth century.

That knowledge, they argue, may have eventually been passed on to Newton himself.

Dr Joseph made the revelations while trawling through obscure Indian papers for a yet to be published third edition of his best selling book 'The Crest of the Peacock: the Non-European Roots of Mathematics' by Princeton University Press.

He said: "The beginnings of modern maths is usually seen as a European achievement but the discoveries in medieval India between the fourteenth and sixteenth centuries have been ignored or forgotten.

"The brilliance of Newton's work at the end of the seventeenth century stands undiminished - especially when it came to the algorithms of calculus.

"But other names from the Kerala School, notably Madhava and Nilakantha, should stand shoulder to shoulder with him as they discovered the other great component of calculus- infinite series.

"There were many reasons why the contribution of the Kerala school has not been acknowledged - a prime reason is neglect of scientific ideas emanating from the Non-European world - a legacy of European colonialism and beyond.

"But there is also little knowledge of the medieval form of the local language of Kerala, Malayalam, in which some of most seminal texts, such as the Yuktibhasa, from much of the documentation of this remarkable mathematics is written."

He added: "For some unfathomable reasons, the standard of evidence required to claim transmission of knowledge from East to West is greater than the standard of evidence required to knowledge from West to East.

"Certainly it's hard to imagine that the West would abandon a 500-year-old tradition of importing knowledge and books from India and the Islamic world.

"But we've found evidence which goes far beyond that: for example, there was plenty of opportunity to collect the information as European Jesuits were present in the area at that time.

"They were learned with a strong background in maths and were well versed in the local languages.

"And there was strong motivation: Pope Gregory XIII set up a committee to look into modernising the Julian calendar.

"On the committee was the German Jesuit astronomer/mathematician Clavius who repeatedly requested information on how people constructed calendars in other parts of the world. The Kerala School was undoubtedly a leading light in this area.

"Similarly there was a rising need for better navigational methods including keeping accurate time on voyages of exploration and large prizes were offered to mathematicians who specialised in astronomy.

"Again, there were many such requests for information across the world from leading Jesuit researchers in Europe. Kerala mathematicians were hugely skilled in this area."

[ICUQB4UQRU]

Wow amazing, I did not know that, thank you for the article.

[moobie]

its a zionist conspiracy to denigrate the great white race, OMG

[Suijen]

I wouldn't be surprised.

[Jagger]

Dr George Gheverghese Joseph already discussed the Keralese discovery of infinite series years ago in his book, Crest of the Peacock: Non-European Roots of Mathematics, which basically covers the mathematical achievements of most non-Western cultures in general (Africans, Babylonians, Egyptians, Indians, Chinese, Meso-Americans, and Muslims).

The Keralese discovery of infinite series has since been recognized in standard mathematical history texts since the turn of the millenium (for example, in Victor Katz's A History of Mathematics: An Introduction). It's already being taught as part of the syllabus in the history of mathematics course I took at my university (but I'm not sure if it's the same for other Western universities?). However, the one thing that many Western scholars are more reluctant to accept is whether European mathematicians were influenced by the Keralese mathematicians, despite the circumstantial evidence.

[froglee]

Oh, my, the Indians predated Newton's discovery? So what's next? Einstein has Indian blood?

This post has been edited by Tenjikuronin: Aug 18 2007, 11:14 AM

[VietGuy7]

Archimedes had already known about infinitesmals over 2000 years ago. From infinitesmals to infinite series is not a big leap. And we should not forget about Zeno's paradoxes.

Also, don't forget that when the Library of Alexander was burned, we lost a lot of ancient Greek mathematics among many other things. It was the Arabs that saved much of the ancient Greek discoveries. Indeed, not all of Euclids books survived. Archimedes himself was a one man think tank, with unbounded creativity. It more reasonable to assume that much of his work, if not most, was lost.

Also, it's hard to know about who influenced who in the ancient world, particularly between the Greeks/Romans and Mideast and India. During the European Dark Ages, the Arabs invented algebra and chemistry, while the Europeans were trying to turn lead into gold. India too was made key contributions throughout the ages. At any rate, it'll be a long time before we fully know what this new revelation from India really means.

Edit: Only about 7 of Sophocles' 100+ plays survived. Sophocles is one of the major contributors to plays and literature in general. The take home point is: A lot was lost.

This post has been edited by VietGuy7: Aug 18 2007, 01:16 AM

[Unity_Asia]

Archimedes had already known about infinitesmals over 2000 years ago. From infinitesmals to infinite series is not a big leap. And we should not forget about Zeno's paradoxes.

Also, don't forget that when the Library of Alexander was burned, we lost a lot of ancient Greek mathematics among many other things. It was the Arabs that saved much of the ancient Greek discoveries. Indeed, not all of Euclids books survived. Archimedes himself was a one man think tank, with unbounded creativity. It more reasonable to assume that much of his work, if not most, was lost.

Also, it's hard to know about who influenced who in the ancient world, particularly between the Greeks/Romans and Mideast and India. During the European Dark Ages, the Arabs invented algebra and chemistry, while the Europeans were trying to turn lead into gold. India too was made key contributions throughout the ages. At any rate, it'll be a long time before we fully know what this new revelation from India really means.

Edit: Only about 7 of Sophocles' 100+ plays survived. Sophocles is one of the major contributors to plays and literature in general. The take home point is: A lot was lost.

Dude, I usually agree with your posts, but this one is ludicrous. an infinitessimal is merely a number that is so much smaller than another number that if you add an infinite amount of them, you will not get the other number. So x+x+x+x.....forever is still less than y, hence x is an infinitessimal. Infinite series, on the other hand, is taking an infinite amount of numbers, and taking their sum. Seriously, infinitessimals is just a concept - a very simple one at that - while infinite series is a way to add an infinite amount of numbers and get the result. How its a simple step from infinitessimals to infinite series is beyond me.
Also, your point about the burning of the Library of Alexander doesn't make much sense, logically. I could say that about any other ancient civilization. If the Nandala University, which housed a massively important library, hadn't been burned down, we might have found evidence of Infinite series being discovered even earlier. I mean, seriously, this has happened to every single civilization that existed. They made enormous discoveries, and wrote many books, most of which were destroyed by some person or group of people. Happened in China, India, wherever. To add to this, ancient Indians weren't known for keeping very good records (unlike the Greeks, who seemed to keep everything), so I can give credit to the Indians on any number of discoveries they didn't actually make using your logic.

[VAMAN]

If the Nandala University, which housed a massively important library, hadn't been burned down, we might have found evidence of Infinite series being discovered even earlier. I mean, seriously, this has happened to every single civilization that existed. They made enormous discoveries, and wrote many books, most of which were destroyed by some person or group of people. Happened in China, India, wherever. To add to this, ancient Indians weren't known for keeping very good records (unlike the Greeks, who seemed to keep everything), so I can give credit to the Indians on any number of discoveries they didn't actually make using your logic.

You are right in this. Actually Nalanda university was destroyed by invading muslim armies. The invasion of muslims and the destruction of various temples and institutions of learning dealt a severe blow to the development of mathematics and sciences espcially in Northern India. Whereas in Southern India where muslim armies couldn't reach that far, the 'Kerala School' of mathematics flourished and contributed vastly to the field of mathematics.

• The great mathematician Aryabhata, studied in Nalanda University. He gave the approximation of Pi = 3.1416. He contributed vastly in Trignomety.
• Very recently the great mathematician Ramanujan. He gave the concept of infinity. He proved that zero divided by zero is infinity. And of course his other exploits in Mathematics are well known.

And of course there were lot more other contributions but I remember these two right now. Also Arabs were greatly influenced by Indian exploits in Mathematics, they made the Europeans aware of these exploits.

This post has been edited by VAMAN: Aug 18 2007, 11:00 PM

[moobie]

Edit: Only about 7 of Sophocles' 100+ plays survived. Sophocles is one of the major contributors to plays and literature in general. The take home point is: A lot was lost.

The same happened in Baghdad and during Qin Shi Huang's rule. Stuff like that is sadly pretty commonplace in history.

This post has been edited by moobie: Aug 18 2007, 04:02 PM

[Jagger]

Archimedes had already known about infinitesmals over 2000 years ago. From infinitesmals to infinite series is not a big leap. And we should not forget about Zeno's paradoxes.

Also, don't forget that when the Library of Alexander was burned, we lost a lot of ancient Greek mathematics among many other things. It was the Arabs that saved much of the ancient Greek discoveries. Indeed, not all of Euclids books survived. Archimedes himself was a one man think tank, with unbounded creativity. It more reasonable to assume that much of his work, if not most, was lost.

Also, it's hard to know about who influenced who in the ancient world, particularly between the Greeks/Romans and Mideast and India. During the European Dark Ages, the Arabs invented algebra and chemistry, while the Europeans were trying to turn lead into gold. India too was made key contributions throughout the ages. At any rate, it'll be a long time before we fully know what this new revelation from India really means.

Edit: Only about 7 of Sophocles' 100+ plays survived. Sophocles is one of the major contributors to plays and literature in general. The take home point is: A lot was lost.

Like Unity just pointed out, infinitesimals and infinite series are two completely different things. The concept of expressing a mathematical function in terms of an infinite series is a major leap in the history of mathematics, and forms the basis of modern mathematical analysis.

It would be more accurate to describe the Library of Alexandria as "Hellenistic" rather than Greek, because Alexandria was in Egypt, not Greece, nor were all the scholars there ethnic Greeks. The works there were all written in Greek because that was the lingua franca of the Mediterrenean at the time, but the scholars in Alexandria were from across the ancient world, including Greeks, Egyptians, Jews, Syrians, Babylonians, and even some Indians.

Like the others have said, libraries were burnt down in almost every civilization (some of these libraries were even larger than the library of Alexandria). A few examples include:

-The libraries of Persepolis were burnt to the ground by Alexander.
-Libraries in Qin China were burnt down by Qin Shi Huang.
-Libraries in Islamic Spain and Syria were destroyed by Christians.
-Libaries in the Abbasid Caliphate (especially Baghdad) and northern China were destroyed by Mongols.
-The libraries of Nalanda and Taxila were destroyed by Muslim Turks.

You are right in this. Actually Nalanda university was destroyed by invading muslim armies. The invasion of muslims and the destruction of various temples and institutions of learning dealt a severe blow to the development of mathematics and sciences espcially in Northern India. Whereas in Southern India where muslim armies couldn't reach that far, the 'Kerala School' of mathematics flourished and contributed vastly to the field of mathematics.

• The great mathematician Aryabhata, studied in Nalanda University. He gave the approximation of Pi = 3.1416. He contributed vastly in Trignomety.
• Very recently the great mathematician Ramanujan. He gave the concept of infinity. He proved that zero divided by zero is infinity. And of course his other exploits in Mathematics are well known.

And of course there were lot more other contributions but I remember these two right now. Also Arabs were greatly influenced by Indian exploits in Mathematics, they made the Europeans aware of these exploits.

Ramanujan was from the 20th century. Don't you mean Brahmagupta from the 7th century?

This post has been edited by Jagger: Aug 21 2007, 10:03 PM

[VAMAN]

Ramanujan was from the 20th century. Don't you mean Brahmagupta from the 7th century?

Lol I know it has nothing to do with ancient mathematics. I could recollect only this at the time of posting. Yes Brahmagupta also made a lot of contribution but I am not much aware of his work. Ancient or modern, Indian scientists and mathematicians have always contributed to it in a big way.

[Jagger]

Lol I know it has nothing to do with ancient mathematics. I could recollect only this at the time of posting. Yes Brahmagupta also made a lot of contribution but I am not much aware of his work. Ancient or modern, Indian scientists and mathematicians have always contributed to it in a big way.

You've already described some of Brahmagupta's 7th century work in your last post. He was the first to give arithmetical rules for zero and infinity. But it was Bhaskara II in the 12th century who first stated that any number divided by 0 is infinity. Ramanujan's 20th century work was something completely different.

This post has been edited by Jagger: Aug 29 2007, 04:08 PM

[VAMAN]

You've already described some of Brahmagupta's 7th century work in your last post. He was the first to give arithmetical rules for zero and infinity. But it was Bhaskara II in the 12th century who first stated that any number divided by 0 is infinity. Ramanujan's 20th century work was something completely different.

Thanks for correcting me I think I got wrong here. It was Bhaskara II who gave the concept of infinity and not Ramanujan. Actually Ramanujam used the concept infinity to solve various mathematical functions and equations in arithmetics and trigonometry also a book 'The Man Who Knew Infinity' about Ramanujan that was in my mind.

Also Brahmagupta was the creator of zero. But Brahmagupta knew nothing about infinity he tried to prove that zero divided by zero is zero which is incorrect.

Brahmagupta then tried to extend arithmetic to include division by zero:-

Positive or negative numbers when divided by zero is a fraction the zero as denominator.
Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
Zero divided by zero is zero.

Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt to extend arithmetic to negative numbers and zero.

http://www-history.mcs.st-andrews.ac.uk/Biographies/Brahmagupta.html

And from wikipedia

Division by zero

Early attempts

The Brahmasphutasiddhanta of Brahmagupta (598-668) is the earliest known text to treat zero as a number in its own right and to define operations involving zero. The author failed, however, in his attempt to explain division by zero: his definition can be easily proven to lead to algebraic absurdities. According to Brahmagupta,

"A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero."

In 830, Mahavira tried unsuccessfully to correct Brahmagupta's mistake in his book in Ganita Sara Samgraha:

"A number remains unchanged when divided by zero."

Bhaskara II tried to solve the problem by defining n/0=infinity. This definition makes some sense, as discussed below, but can lead to paradoxes if not treated carefully. These paradoxes were not treated until modern times.

http://en.wikipedia.org/wiki/Division_by_zero

This post has been edited by VAMAN: Aug 31 2007, 01:20 AM

[VietGuy7]

Infinite series, on the other hand, is taking an infinite amount of numbers, and taking their sum. Seriously, infinitessimals is just a concept - a very simple one at that - while infinite series is a way to add an infinite amount of numbers and get the result. How its a simple step from infinitessimals to infinite series is beyond me.

I think you just answered your own question before you posed the question... Below is the portion that you actually wrote first:

an infinitessimal is merely a number that is so much smaller than another number that if you add an infinite amount of them, you will not get the other number. So x+x+x+x.....forever is still less than y, hence x is an infinitessimal.

Infinite series themselves are not interesting unless they converge. And the article itself says nothing as to whether the Kerela school knew about convergence. Of course, divergent series are quite uninteresting to begin with.

To add to this, ancient Indians weren't known for keeping very good records (unlike the Greeks, who seemed to keep everything), so I can give credit to the Indians on any number of discoveries they didn't actually make using your logic.

Everything we have from the Greeks comes from the Arabs. The Chinese/Confucians were by far the most meticulous record keepers of ancient times--certaily much more so than the ancient Greeks.

so I can give credit to the Indians on any number of discoveries they didn't actually make using your logic.

My logic was based on the fact that it doesn't take much of a leap from infinitesmals to the concept of convergence (of infinite series), which is without question the most interesting thing about infinite series. Think about it.

The addition of an infinite number of infinitesmals will converge.
The addition of a infinite convergent series will converge.

It's just a matter of replacing the infinitesmal with a number, which is in turn based on some formula.

Now they say that the three greatest mathematicians of all time are:

Archimedes, Newton & Gauss. Do you have any idea of the kind of brain power we're talking about here?!? If you think it is beyond Archimedes' ability to make the leap from infinitesmals to infinite series, then you are quite mistaken. Archimedes is known for so much stuff--an increadible amount of stuff. And, like Sophocles and Aeschylus, I'm pretty damn sure we don't have everything he had ever come up with.

We're not talking about going from the Pythagorean Theorem to Atiya-Singer Index Formula here. The leap from infinitesmals and infinite series is not that hard to make.

P.S. Methinks I should look up Newton's calculus work, for I now am wondering if it is the case that Newton read Archimedes and simply made that little leap of logic I spoke of above. I'm sure you know that Newton made a ton of leaps of logic, far more difficult than fixing a number to an infinitesmal, when he came up with calculus. And I'm sure you know that Newton invented calculus just to prove his even more impressive theories in physics. Like I've been suggesting: people with a lot of brain power can make leaps of logic, leaps of innovation quite easily.

P.P.S. Sophocles wrote a total of at least 123 plays. Only 7 survive. These seven alone make him one of the greatest playwright of al time. Same goes for Aeschylus, the father of tragedy, who wrote at least 70 plays, but only 7 have survived. Aeschylus is even a greater dramatist than Sophocles.

P.P.P.S. Shakespeare only wrote 36 plays--and we have ALL of them. He is universally hailed as the greatest writer of all time. I'd say only half of them are really that important: Hamlet, Lear, Othello, Macbeth, Henry V, The Tempest, A Midsummer Nights Dream... Hmm, wonder how Shakespeare would be ranked if only 1/10 or 7/123 of his plays survived?

P.P.P.P.S Of course, history is based on actual records an not lost records. And in this sense, this new discovery does warrant that we give credit to the Indians. Happy?

[VietGuy7]

Like Unity just pointed out, infinitesimals and infinite series are two completely different things. The concept of expressing a mathematical function in terms of an infinite series is a major leap in the history of mathematics, and forms the basis of modern mathematical analysis.

A "major leap"? Sure. But not to the likes of Newton. Just read his Principia and scour it for new ideas. You'll find a mountain of major leaps. Archimedes was no slouch, if you didn't know.

It would be more accurate to describe the Library of Alexandria as "Hellenistic" rather than Greek, because Alexandria was in Egypt, not Greece, nor were all the scholars there ethnic Greeks. The works there were all written in Greek because that was the lingua franca of the Mediterrenean at the time, but the scholars in Alexandria were from across the ancient world, including Greeks, Egyptians, Jews, Syrians, Babylonians, and even some Indians.

Again, my point was really that everything we have from the Greeks and much of that of the Romans too, comes from the Arabs.

Like the others have said, libraries were burnt down in almost every civilization (some of these libraries were even larger than the library of Alexandria). A few examples include:

-The libraries of Persepolis were burnt to the ground by Alexander.
-Libraries in Qin China were burnt down by Qin Shi Huang.
-Libraries in Islamic Spain and Syria were destroyed by Christians.
-Libaries in the Abbasid Caliphate (especially Baghdad) and northern China were destroyed by Mongols.
-The libraries of Nalanda and Taxila were destroyed by Muslim Turks.
Ramanujan was from the 20th century. Don't you mean Brahmagupta from the 7th century?

Well, I'm Vietnamese, let me tell you that Vietnamese are still pissed off as Hell about the Ming carting away and either burning or losing our records in the 1400's. Many Vietnamese claim we invented this or that but we can't prove any of it since the Chinese took our records.

Having said that, I'm pretty sure it safe to say that our libraries were not filled with the same calibre of work as was done by Archimedes, Aeschylus, et al. We mostly copied the ancient Chinese. The ancient Greeks certainly outdid the Ancient Chinese in thought. Heck, the Ancient Chinese got much of their own philosophy, religion and so on from India.

But again, history is based on actual evidence, not lost evidence. So the desi's get full credit. And I was wrong.

[Jagger]

I think you just answered your own question before you posed the question... Below is the portion that you actually wrote first:

Infinite series themselves are not interesting unless they converge. And the article itself says nothing as to whether the Kerela school knew about convergence. Of course, divergent series are quite uninteresting to begin with.

Everything we have from the Greeks comes from the Arabs. The Chinese/Confucians were by far the most meticulous record keepers of ancient times--certaily much more so than the ancient Greeks.

My logic was based on the fact that it doesn't take much of a leap from infinitesmals to the concept of convergence (of infinite series), which is without question the most interesting thing about infinite series. Think about it.

The addition of an infinite number of infinitesmals will converge.
The addition of a infinite convergent series will converge.

It's just a matter of replacing the infinitesmal with a number, which is in turn based on some formula.

Now they say that the three greatest mathematicians of all time are:

Archimedes, Newton & Gauss. Do you have any idea of the kind of brain power we're talking about here?!? If you think it is beyond Archimedes' ability to make the leap from infinitesmals to infinite series, then you are quite mistaken. Archimedes is known for so much stuff--an increadible amount of stuff. And, like Sophocles and Aeschylus, I'm pretty damn sure we don't have everything he had ever come up with.

We're not talking about going from the Pythagorean Theorem to Atiya-Singer Index Formula here. The leap from infinitesmals and infinite series is not that hard to make.

P.S. Methinks I should look up Newton's calculus work, for I now am wondering if it is the case that Newton read Archimedes and simply made that little leap of logic I spoke of above. I'm sure you know that Newton made a ton of leaps of logic, far more difficult than fixing a number to an infinitesmal, when he came up with calculus. And I'm sure you know that Newton invented calculus just to prove his even more impressive theories in physics. Like I've been suggesting: people with a lot of brain power can make leaps of logic, leaps of innovation quite easily.

It's not as simple as you claim. The concept of expressing a mathematical function (such as pi or the trigonometric functions) as a convergent infinite series expansion had never been thought of before the 14th century. If it was so simple, it wouldn't have taken over 1600 years (from around 200 BC to 1400 AD) to make that leap from infinitesimals in the first place. Almost every historian of mathematics is aware of the significance of this leap, and I have not heard of one scholar ever claiming it was a small leap. Even if we do assume it was a "small leap" as you claim, the Kerala mathematicians came far closer to inventing the calculus than just discovering the infinite series.

In the 14th century, Madhava of Sangamagrama not only discovered the concept of the convergent infinite series, but he also discovered the power series expansion for pi, the Taylor series expansions for the trigonometric functions of sine, cosine, tangent, and arctangent, the Taylor series approximations for sine and cosine, the test of convergence for infinite series, the analysis of infinite continued fractions, etc. In the 12th century, Bhaskara II discovered the concept of the derivative and differential coefficient, solved the first differential equation, and first stated "Rolle's theorem". In the 15th century, Parameshvara discovered the mean value theorem. In the 16th century, Jyeshtadeva gave proofs for the infinite series expansions of pi and the trigonometric functions, solved integrals, and discovered the method of term-by-term integration. And there were a number of other discoveries. All of these were compiled into Jyeshtadeva's text "Yuktibhasa", which is regarded as the first calculus text.

They only thing that was left to do to invent the calculus was to unify the theories of the differential and the integral along with mathematical analysis, which is what Newton and Leibniz became famous for.

P.S. Yes, it was beyond Archimedes' ability to make the leap from infinitesimals to the infinite series expansions of mathematical functions. The mathematical knowledge in his time was insufficient to make such a leap, regardless of his ability.

P.P.S. Sophocles wrote a total of at least 123 plays. Only 7 survive. These seven alone make him one of the greatest playwright of al time. Same goes for Aeschylus, the father of tragedy, who wrote at least 70 plays, but only 7 have survived. Aeschylus is even a greater dramatist than Sophocles.

P.P.P.S. Shakespeare only wrote 36 plays--and we have ALL of them. He is universally hailed as the greatest writer of all time. I'd say only half of them are really that important: Hamlet, Lear, Othello, Macbeth, Henry V, The Tempest, A Midsummer Nights Dream... Hmm, wonder how Shakespeare would be ranked if only 1/10 or 7/123 of his plays survived?

How exactly is drama relevant to mathematics?

A "major leap"? Sure. But not to the likes of Newton. Just read his Principia and scour it for new ideas. You'll find a mountain of major leaps. Archimedes was no slouch, if you didn't know.

Again, my point was really that everything we have from the Greeks and much of that of the Romans too, comes from the Arabs.

Well, I'm Vietnamese, let me tell you that Vietnamese are still pissed off as Hell about the Ming carting away and either burning or losing our records in the 1400's. Many Vietnamese claim we invented this or that but we can't prove any of it since the Chinese took our records.

Having said that, I'm pretty sure it safe to say that our libraries were not filled with the same calibre of work as was done by Archimedes, Aeschylus, et al. We mostly copied the ancient Chinese. The ancient Greeks certainly outdid the Ancient Chinese in thought. Heck, the Ancient Chinese got much of their own philosophy, religion and so on from India.

No one is doubting the genius of Newton, but there are a handful of other universal geniuses who can rival his achievements.

Well actually, most of the surviving ancient Greek works were preserved by both the Arabic Muslims and the Byzantine Greeks, but the Latin Europeans preferred the Arabic versions because of the commentaries that came with them.

Maybe not in philosophy, but the Chinese certainly outdid the Greeks in technology.

This post has been edited by Jagger: Sep 16 2007, 01:35 AM

[VietGuy7]

^ Are convergent series all about pi?!? You mean much simpler convergent series don't exist?

You need to read Newton and Archimedes before you say that they could not have made such "major leaps". Newton's Principia is full of "major" leaps, which were much, much, much bigger than the one going from infinitesmals to infinite series. Archimedes was no different.

While you're at it, read Ramanujan too. He did a Hell of a lot more than make one or two major leaps. He probably equalled Newton--it's only that his work was so much narrower in focus.

This post has been edited by VietGuy7: Sep 16 2007, 12:17 PM

[VietGuy7]

Maybe not in philosophy, but the Chinese certainly outdid the Greeks in technology.

The Chinese also outnumbered the Greeks. BIG TIME! Still do too.

P.S. Again, a lot of the Greek as well as Roman stuff was lost. Further, the Greeks got much of their inspiration from the Egyptians and no doubt their nemesis the Persians.

How exactly is drama relevant to mathematics?

I'm talking about lost records here. How is it that the vast majority of the works of Sophocles (95%) and Aeschylus (90%) was lost, but all the works of ancient Greek mathematicians survived?

This post has been edited by VietGuy7: Sep 16 2007, 12:15 PM

[Jagger]

^ Are convergent series all about pi?!? You mean much simpler convergent series don't exist?

You need to read Newton and Archimedes before you say that they could not have made such "major leaps". Newton's Principia is full of "major" leaps, which were much, much, much bigger than the one going from infinitesmals to infinite series. Archimedes was no different.

While you're at it, read Ramanujan too. He did a Hell of a lot more than make one or two major leaps. He probably equalled Newton--it's only that his work was so much narrower in focus.

The first convergent infinite series that was discovered was for pi, but this would not have been possible without the power series for the arctangent, which itself would require knowledge of the power series of sine and cosine, which itself requires knowledge of all six trigonometric functions and the sum formulas for integral powers to find the area under a curve, which itself requires knowledge of mathematical induction and the sum formulas for the squares, cubes and fourth powers, which itself requires even more mathematical knowledge. This is why Archimedes, despite being a mathematical genius, was unable to discover the infinite series, simply because all of this mathematical knowledge did not exist in his time. Making a jump from infinitesimals to convergent infinite series might now seem simple on the surface, but it required a lot of knowledge from different fields of mathematics before it was possible to make that leap. In a sense, Archimedes' use of infinitesimals was not such a big leap from what the Egyptians did in the Moscow papyrus (their solution to the volume of a truncated pyramid marked the beginning of integral calculus), but more knowledge of geometry was required before it was possible to make that leap.

Also, like I already explained in my previous post, the Kerala mathematicians did not simply discover the infinite series, but they went far beyond that, and even came close to inventing the calculus. All that they lacked was the unification of the differential, integral and infinite series into a whole, and this is where Newton succeeded. Like I said before, no one is doubting Newton's ability (or Archimedes' ability). Newton was a universal genius, and even the article agrees with this, but his ingenuity does not simply lie in making major discoveries, but in how he applied those discoveries. In mathematics, many ideas on the calculus were already known before him (by earlier European mathematicians and Indian mathematicians before them), but what Newton achieved was unifying all these ideas into a whole, now known as the calculus. Similarly in physics, he unified different physical and astronomical theories already known before him (by earlier European scientists and Muslim scientists before them) into his laws of motion and law of gravity. What he is most famous for was making a connection between different theories and ideas and unifying them into a whole.

Ramanujan was a mathematical genius who specialized in the field of mathematics, while Newton was a universal genius who contributed to different fields of knowledge. Although Newton is best known for his works on physics and mathematics, his works on alchemy and theology have often been ignored, probably because of their occult content.

The Chinese also outnumbered the Greeks. BIG TIME! Still do too.

P.S. Again, a lot of the Greek as well as Roman stuff was lost. Further, the Greeks got much of their inspiration from the Egyptians and no doubt their nemesis the Persians.

The Greeks made up for the population difference with all of the knowledge they borrowed from the Babylonians, Egyptians, Phoenicians, Persians and Indians. The Chinese did most of their work independantly, although they did borrow ideas from the Indians (and possibly the Babylonians) but to a lesser extent. Also, the Greek population expanded significantly after Alexander's conquest of the Persian Empire. There were many Egyptians, Babylonians, Phoenicians and Persians who were Hellenized at the time and adopted Greek language and culture. A significant number of the so-called "Greek" mathematicians and philosophers during Hellenistic and Roman times were in fact Hellenized Egyptians.

P.S. The Chinese lost a lot more of their works during Qin Shi Huang's book-burning incident. The same goes for the Indians during the destruction of Nalanda University, and the Muslims during the Crusades and Mongol invasions.

I'm talking about lost records here. How is it that the vast majority of the works of Sophocles (95%) and Aeschylus (90%) was lost, but all the works of ancient Greek mathematicians survived?

That's probably because the medieval Muslims had very little interest in Greek drama or poetry. They were more interested in translating Greek works on mathematics and philosophy instead.

This post has been edited by Jagger: Sep 16 2007, 04:02 PM